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Properties

Label 2-231-231.65-c1-0-11
Degree $2$
Conductor $231$
Sign $0.923 + 0.383i$
Analytic cond. $1.84454$
Root an. cond. $1.35814$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.36 + 2.36i)2-s + (−0.516 + 1.65i)3-s + (−2.72 − 4.72i)4-s + (−1.96 − 1.13i)5-s + (−3.20 − 3.47i)6-s + (−1.90 + 1.83i)7-s + 9.44·8-s + (−2.46 − 1.70i)9-s + (5.37 − 3.10i)10-s + (3.29 − 0.415i)11-s + (9.22 − 2.07i)12-s − 0.322i·13-s + (−1.74 − 7.01i)14-s + (2.89 − 2.66i)15-s + (−7.43 + 12.8i)16-s + (−1.73 − 2.99i)17-s + ⋯
L(s)  = 1  + (−0.965 + 1.67i)2-s + (−0.298 + 0.954i)3-s + (−1.36 − 2.36i)4-s + (−0.880 − 0.508i)5-s + (−1.30 − 1.42i)6-s + (−0.719 + 0.694i)7-s + 3.33·8-s + (−0.822 − 0.569i)9-s + (1.69 − 0.981i)10-s + (0.992 − 0.125i)11-s + (2.66 − 0.597i)12-s − 0.0894i·13-s + (−0.465 − 1.87i)14-s + (0.747 − 0.688i)15-s + (−1.85 + 3.21i)16-s + (−0.419 − 0.727i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 + 0.383i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.923 + 0.383i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(231\)    =    \(3 \cdot 7 \cdot 11\)
Sign: $0.923 + 0.383i$
Analytic conductor: \(1.84454\)
Root analytic conductor: \(1.35814\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{231} (65, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 231,\ (\ :1/2),\ 0.923 + 0.383i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0982696 - 0.0196016i\)
\(L(\frac12)\) \(\approx\) \(0.0982696 - 0.0196016i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.516 - 1.65i)T \)
7 \( 1 + (1.90 - 1.83i)T \)
11 \( 1 + (-3.29 + 0.415i)T \)
good2 \( 1 + (1.36 - 2.36i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + (1.96 + 1.13i)T + (2.5 + 4.33i)T^{2} \)
13 \( 1 + 0.322iT - 13T^{2} \)
17 \( 1 + (1.73 + 2.99i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.16 + 1.82i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.67 - 1.54i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 6.42T + 29T^{2} \)
31 \( 1 + (2.42 + 4.19i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.690 - 1.19i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 3.42T + 41T^{2} \)
43 \( 1 + 5.82iT - 43T^{2} \)
47 \( 1 + (2.37 + 1.37i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-0.824 + 0.475i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (9.17 - 5.29i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.0609 - 0.0351i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.262 - 0.455i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 2.71iT - 71T^{2} \)
73 \( 1 + (3.18 - 1.83i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-4.08 - 2.35i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 2.50T + 83T^{2} \)
89 \( 1 + (-2.19 - 1.26i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 14.2T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.91060349941935197661999142428, −10.88921630355548774110740639594, −9.593922164922834459773760348840, −9.101508058047700907766504344178, −8.412091304127729363107214484009, −7.08571665456985581369687287045, −6.12415387719777283655353214241, −5.13310188968577826383256426617, −4.02164355636109692866616395244, −0.12043947225080223206370882098, 1.60222933175274712921865318158, 3.21821268988863344885083647156, 4.13619894058469122800118444521, 6.65842994770941152175372348285, 7.51462720874930100059297507005, 8.477761876297203265753255524640, 9.492294346469131522253245407807, 10.77443885615062282103814849103, 11.11055694435165246755749152052, 12.12714558981212362011984303308

Graph of the $Z$-function along the critical line